Dale McLerran

2006-01-24 19:49:35 UTC

Hello

With ODS graphics in PROC MIXED, SAS can produce studentized,

marginal, and raw residuals, each can be conditional or marginal.

How do these relate to the assumptions of the model?

y = Xbeta + Zgamma + epsilon

E(gamma) = E(epsilon) = 0

V(gamma) = G

V(epsilon) = R

I understand that studentization and Pearsonization (if that's the

word) are ways to standardize the raw numbers;

my question is more about the conditional vs. the marginal. I see

that (on p. 2764 in the SAS STAT manuals)

r marginal _i = Y_i -x'_i*betahat

r conditional_i = r_mi - z'_i*gammahat

this seems to me to suggest that the marginal residuals are somehow

about G, and the conditional residuals about R.....but I am not at

all sure.....

Thanks as always

Peter

Peter L. Flom, PhD

Peter,With ODS graphics in PROC MIXED, SAS can produce studentized,

marginal, and raw residuals, each can be conditional or marginal.

How do these relate to the assumptions of the model?

y = Xbeta + Zgamma + epsilon

E(gamma) = E(epsilon) = 0

V(gamma) = G

V(epsilon) = R

I understand that studentization and Pearsonization (if that's the

word) are ways to standardize the raw numbers;

my question is more about the conditional vs. the marginal. I see

that (on p. 2764 in the SAS STAT manuals)

r marginal _i = Y_i -x'_i*betahat

r conditional_i = r_mi - z'_i*gammahat

this seems to me to suggest that the marginal residuals are somehow

about G, and the conditional residuals about R.....but I am not at

all sure.....

Thanks as always

Peter

Peter L. Flom, PhD

The conditional residuals are obtained as

Rc_i = Y_i - E(Y|x,z)

= Y_i - (x'_i*betahat + z'_i*gammahat)

The marginal residuals are obtained as

Rm_i = Y_i - E(Y|x)

= Y_i - x'_i*betahat

Suppose that you have a new cluster/subject that was not

part of your estimation model. Thus, you have no estimate

of the random effects which pertain to this new subject.

You cannot compute z'_i*gammahat, so you cannot compute

subject-specific (conditional) residuals. You can compute

the marginal residuals (assuming that there are no missing

values in the vector x_i).

The marginal residuals, then, are residuals that have a

distribution which is quite nearly the population distribution

of residuals for your fixed-effect model. If you went out

into the world armed with your mixed model, you would only

be able to apply the fixed effect portion of the model.

The marginal residuals represent just how discrepent the

fixed effect model would be over the population of subjects

in the absence of knowledge of subject-specific effects.

(Well, this is not quite true, because the residuals are

obtained so as to achieve best fit in the observed data.)

Note that the marginal residuals do not represent how

discrepent the fixed effect model would be for a single

subject with unspecified subject-specific effects. The

residuals for a single subject with unspecified subject-

specific effects will tend toward all positive or all

negative due to the exclusion of the subject-specific

effects. Thus, the marginal residuals will be biased

for a given subject, with bias due to the subject-specific

effects which are excluded from the marginal model.

The conditional residuals provide an indication of how well

your model estimates the response WHEN YOU KNOW SUBJECT-

SPECIFIC EFFECTS. The conditional residuals may be better

than the marginal residuals for model diagnostic purposes -

evaluating linearity and heteroskedasticity of the response

as a function of the variables included in your model.

That is, the conditional residuals may be useful for

constructing your model. However, the conditonal residuals

will not provide a clear picture of how well the model

works in the population.

Dale

---------------------------------------

Dale McLerran

Fred Hutchinson Cancer Research Center

mailto: ***@NO_SPAMfhcrc.org

Ph: (206) 667-2926

Fax: (206) 667-5977

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